Tuesday, July 29, 2014

The Sleeping Beauty problem

This is a simple problem of probabilities, involving tossing a coin. But for some reason, it seems to be no agreement about its solution.

Consider the following experiment:

- On Sunday, put Sleeping Beauty to sleep.

- Toss a fair coin.

- We are interested to ask Sleeping Beauty the question

Q. What is your belief now for the proposition that the coin landed heads?

- If the coin comes up heads, wake up Sleeping Beauty on Monday and ask her the question. Then drug her to forget that awakening.

- If the coin comes up tails, wake up Sleeping Beauty both on Monday and Tuesday and ask her the question. Each time drug her to forget that awakening.

- In both cases, don't forget to wake her on Wednesday and end the experiment.

If you have trouble convincing a Beauty to let you put her to sleep and drug her, you can try your luck with people who already have very short memory, like Lucy Whitmore from "50 First Dates", Leonard from "Memento", Allie from The Notebook, or Dory from "Finding Nemo". Or you can make the experiment with Dory from "Finding Nemo".

Those thinking they know the answer are mainly in one of two camps: halfers, who think she should answer 1/2, and thirdirs, who think she should answer 1/3. Thirdirs say that when Beauty is waken and interviewed, she thinks she
can be in one of three situations. Since only in one of the cases the
coin turned up heads, the answer must be 1/3. Halfers say that this answer is wrong, being probably caused by drug abuse, and since the coin is fair, the answer should be 1/2. There is nothing that can provide new information to Sleeping Beauty, so this answer should remain 1/2.

I will not detail here the debates still ongoing on the net, and the articles which are written about this. I just want to explain why I think that this debate is based on different understandings of the question.

Another experiment

Consider the following experiment.

- Prepare a large box, in which you can put apples and oranges, without seeing its content.

- Toss a fair coin.

- If the coin comes up heads, put one orange in the box.

- If the coin comes up tails, put two apples in the box.

- Repeat this many times.

- At the end, randomly extract a fruit from the box. Unless the experiment took too long, the fruits are not yet rotten, so you can extract a fruit.

- Then answer the following questions:

1. What is the probability that the fruit you will extract was introduced after the coin landed heads?

2. What fraction of the total times the coin was tossed, it landed heads?

The answer to question 1 is of course 1/3, because 1/3 of the fruits are oranges, and oranges were placed in the box when the coin landed heads.

The answer to question 2 is of course 1/2, because the coin is supposed to be fair.

My claim is that thirdirs were actually answering question 1, and halfers were answering question 2.

The question Sleeping Beauty was asked can be seen as equivalent to both question 1 and question 2.

To see how it can be seen as equivalent to question 1, consider a combination of the two experiments. Say that Sleeping Beauty is not only asked the question, but also it is given a fruit to put in the box. If the coin landed heads, she will receive an orange, and if it landed tails, she will receive an apple. She will put them in the box, then she will be put to sleep and forget about the awakening. Say the experiment is ran a large number of times. At the end, she can just count the fruits, and she will find that 1/3 of them are indeed oranges, so she will know that indeed the answer to the question is 1/3. Asking her about her belief that the coin landed heads that time is the same as asking her about her belief that she will receive an orange.

It is true that for every time the coin landed tails, she gets two apples, while every time it landed heads, she gets only one orange. This is why some tend to understand the question as being actually question 2.

Removing the confusion

So the dispute between thirdirs and halfers is due to the fact that they interpreted the question differently, and consequently answered different questions.

Instead of asking Sleeping Beauty the question as originally stated, we could just ask her two questions:

Q1. What is your belief that this awakening occurred following an event in which the coin landed heads?

Q2. What is your belief that when the coin was tossed, it landed heads?